One dimensional fractional frequency Sumudu transform by inverse α−difference operator

In this paper, we define fractional frequency Sumudu transform by inverse α−difference operator. Here we present certain new results on Sumudu transform of polynomial factorial, trigonometric and geometric functions using shift value. Finally, we provide the relation between convolution product and fractional Sumudu transform of polynomial and exponential function. Numerical results are verified and analysed the outcomes by graphs.


Introduction
There are several integral transforms such as the Laplace, Millen, Hankel and Fourier transforms that are used to solve differential equations which apper in many fields of science and engineering. In the early 1990's, Watugala [9,10] introduced the Sumudu transform and applied it to solve ordinary differential equations. Watugala's work was followed by Weerakoon who introduced the complex inversion formula for the Sumudu transform [11,12]. The fundamental properties of this transform, which is thought to be an alternative to the Laplace transform were then established in many articles [13,14]. The Sumudu transform is defined over the set of functions by Also, there is an bridge between Laplace transform and Sumudu transform which has many applications in applied sciences. Moreover, some properties of Sumudu transform makes it more advantageous than the Sumudu transform of a Heaviside step function is a also Heaviside step function in the transformed domain; −St n = n!u n ; − lim Recently, it was proved that by using the Sumudu transform, one can transform the two dimensional transport equation into a Fredholm integral equation [18]. In [17], the authors applied the Sumudu transform to fractional differential equations which have many applications in the fields of science (see [19] and the references therein).
Begin with classical definition of Laplace transform an arbitrary time scales, the concept of the h−Laplace and consequently the discrete Laplace transformed were specified in [15]. It was initiated by Stefan Hilger [16]. This theory is a tool that unifies the theories of continuous and discrete time systems. It is the subject of recent studies on many different fields in which dynamic process can be described with discrete or continuous models. The recent applications of fractional Laplace transform using difference equation are found in [1,2,20,21,22].
In this research article, we proposed a new type of Sumudu transform with shift value and the properties are discussed. Several results are derived to validate the definition and also the relation between convolution product and sumudu transform are played a vital role using α-difference operator.

Preliminaries
In this section, we present basic theory of the h−difference operator ∆ h . The polynomial factorial is defined t , h > 0 for non-negative integer m and using Stirling numbers of first kind s m r and second kind S m r , the relation between polynomial and polynomial factorials are given by, Definition 2.1 Let u(t), t ∈ [0, ∞), be a real or complex valued function and h > 0 be a fixed shift value. Then, the α(h)−difference operator ∆ α(h) on u(t) is defined as and its infinite h− difference sum is defined by Remark 2.2 When α = h = 1 in (4) we get ∆u(t) = u(t + 1) − u(t).

Lemma 2.3
Let u(t) and v(t) are the two real valued functions defined on (−∞, ∞) and if ∆ α(h) v(t) = u(t), then the finite inverse principle law is given by  Lemma 2.4 [22] Let h > 0 and u(t), w(t) are real valued bounded functions. Then .

Alpha Fractional Frequency Sumudu Transform
In this section, we derive several results and identities on fractional frequency Sumudu transform of polynomial factorial, trigonometric and geometric functions.
− t τ ν = 0, then the fractional frequency Sumudu transform with tuning factor α is defined as Proof. The proof of (13) follows by multiplying 1 τ and applying limits from 0 to ∞ in (10) and (11). The following example is the numerical verification of Theorem Proof. From the definition of Sumudu transform, we have Apply ∆ −1 α(h) on both sides we get, In the similar manner, we arrives , DOI : http://doi.org/10.26524/cm73 which completes the proof of (15).

Convolution Product and Fractional Sumudu Transforms
In this section, we defined convolution product with Fractional Sumudu transforms.
The following definitions are motivated using α(h)−difference operator.
The following Figure 3

Conclusion
In this research article, we introduced and derived results on fractional frequency Sumudu transform with shift values and using α−difference operator. We believe that this transform is an alternative in the field of difference equations. The more advantage of this research is when ν = 1, α = 1 and h → 0, we get the same results on classical Sumudu transform which is existing in the literature.